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Theorem elcncf1di 12749
Description: Membership in the set of continuous complex functions from 
A to  B. (Contributed by Paul Chapman, 26-Nov-2007.)
Hypotheses
Ref Expression
elcncf1d.1  |-  ( ph  ->  F : A --> B )
elcncf1d.2  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ ) )
elcncf1d.3  |-  ( ph  ->  ( ( ( x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )  ->  (
( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
Assertion
Ref Expression
elcncf1di  |-  ( ph  ->  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) ) )
Distinct variable groups:    x, w, y, A    w, B, x, y    w, F, x, y    ph, w, x, y   
w, Z
Allowed substitution hints:    Z( x, y)

Proof of Theorem elcncf1di
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elcncf1d.1 . . 3  |-  ( ph  ->  F : A --> B )
2 elcncf1d.2 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ ) )
32imp 123 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  RR+ ) )  ->  Z  e.  RR+ )
4 an32 551 . . . . . . . . 9  |-  ( ( ( x  e.  A  /\  w  e.  A
)  /\  y  e.  RR+ )  <->  ( ( x  e.  A  /\  y  e.  RR+ )  /\  w  e.  A ) )
54anbi2i 452 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )
)  <->  ( ph  /\  ( ( x  e.  A  /\  y  e.  RR+ )  /\  w  e.  A ) ) )
6 anass 398 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  RR+ ) )  /\  w  e.  A
)  <->  ( ph  /\  ( ( x  e.  A  /\  y  e.  RR+ )  /\  w  e.  A ) ) )
75, 6bitr4i 186 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )
)  <->  ( ( ph  /\  ( x  e.  A  /\  y  e.  RR+ )
)  /\  w  e.  A ) )
8 elcncf1d.3 . . . . . . . 8  |-  ( ph  ->  ( ( ( x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )  ->  (
( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
98imp 123 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )
)  ->  ( ( abs `  ( x  -  w ) )  < 
Z  ->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
) )
107, 9sylbir 134 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  RR+ ) )  /\  w  e.  A
)  ->  ( ( abs `  ( x  -  w ) )  < 
Z  ->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
) )
1110ralrimiva 2505 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  RR+ ) )  ->  A. w  e.  A  ( ( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) )
12 breq2 3933 . . . . . 6  |-  ( z  =  Z  ->  (
( abs `  (
x  -  w ) )  <  z  <->  ( abs `  ( x  -  w
) )  <  Z
) )
1312rspceaimv 2797 . . . . 5  |-  ( ( Z  e.  RR+  /\  A. w  e.  A  (
( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) )  ->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w
) )  <  z  ->  ( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) )
143, 11, 13syl2anc 408 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  RR+ ) )  ->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w ) )  < 
z  ->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
) )
1514ralrimivva 2514 . . 3  |-  ( ph  ->  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w
) )  <  z  ->  ( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) )
161, 15jca 304 . 2  |-  ( ph  ->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
17 elcncf 12743 . 2  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w
) )  <  z  ->  ( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) ) )
1816, 17syl5ibrcom 156 1  |-  ( ph  ->  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1480   A.wral 2416   E.wrex 2417    C_ wss 3071   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   CCcc 7630    < clt 7812    - cmin 7945   RR+crp 9453   abscabs 10781   -cn->ccncf 12740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-map 6544  df-cncf 12741
This theorem is referenced by:  elcncf1ii  12750  cncfmptc  12765  cncfmptid  12766  addccncf  12769  negcncf  12771
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